A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue

This paper is concerned with the rather broad issue of the impact of abrupt changes (such as isolated roughness, gaps and local suctions) on boundary-layer transition. To fix the idea, we consider the influence of a two-dimensional localized hump (or indentation) on an oncoming Tollmien-Schlichting (T-S) wave. We show that when the length scale of the former is comparable with the characteristic wavelength of the latter, the key physical mechanism to affect transition is through scattering of T-S waves by the roughness-induced mean-flow distortion. An appropriate mathematical theory, consisting of the boundary-value problem governing the local scattering, is formulated based on triple deck formalism. The transmission coefficient, defined as the ratio of the amplitude of the T-S wave downstream of the roughness to that upstream, serves to characterize the impact on transition. The transmission coefficient appears as the eigenvalue of the discretized boundary-value problem. The latter is solved numerically, and the dependence of the eigenvalue on the height and width of the roughness and the frequency of the T-S wave is investigated. For a roughness element without causing separation, the transmission coefficient is found to be approximately 1.5 for typical frequencies, indicating a moderate but appreciable destabilizing effect. For a roughness causing incipient separation, the transmission coefficient can be as large as O(10), suggesting that immediate transition may take place at the roughness site. A roughness element with a fixed height produces the strongest impact when its width is comparable with the T-S wavelength, in which case the traditional linear stability theory is invalid. The latter however holds approximately when the roughness width is sufficiently large. By studying the two hump case, a criterion when two roughness elements can be regarded as being isolated is suggested. The transmission coefficient can be converted to an equivalent N-factor increment, by making use of which the e(N)-method can be extended to predict transition in the presence of multiple roughness elements.